This thesis investigates a new proposal for a privileged ground state of a free scalar quantum field in arbitrary regions of spacetime. This Sorkin-Johnston (SJ) state, implicit in work by S. Johnston on quantum field theory on causal sets, is defined solely in terms of the spacetime causal structure and is unique in any globally hyperbolic spacetime region. The first part of the thesis contains an analysis of the simplest possible setting: a flat two-dimensional causal interval. The simplicity of the setup makes analytic calculations tractable and allows for some general features of the state to be better understood. The second part deals with an investigation of the SJ state in de Sitter space. It turns out to be possible to construct the state explicitly using limiting procedures, which provides further interesting insights. In particular, the state is found to depend on the spacetime dimension, field mass, and on the choice of subregion, differing in many cases from the usual 'Bunch-Davies' vacuum. The formalism does not select a unique state in spacetimes that are not globally hyperbolic, which include, among others, spacetimes exhibiting spatial topology change. These are relevant in the context of quantum gravity and in relation to the old question as to whether violent spacetime curvature fluctuations at Planckian scales can lead to changes in spatial topology, or whether such transitions are unphysical. Some efforts to understand the SJ state in the topology-changing two-dimensional 'trousers' spacetime are discussed in the final part of the thesis.